Simplicial map neural networks (SMNNs) are topology-based neural networks with interesting properties such as universal approximation ability and robustness to adversarial examples under appropriate conditions. However, SMNNs present some bottlenecks for their possible application in high-dimensional datasets. First, SMNNs have precomputed fixed weight and no SMNN training process has been defined so far, so they lack generalization ability. Second, SMNNs require the construction of a convex polytope surrounding the input dataset. In this paper, we overcome these issues by proposing an SMNN training procedure based on a support subset of the given dataset and replacing the construction of the convex polytope by a method based on projections to a hypersphere. In addition, the explainability capacity of SMNNs and effective implementation are also newly introduced in this paper.

中文翻译：

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可训练和可解释的单纯图神经网络

简单映射神经网络（SMNN）是基于拓扑的神经网络，具有有趣的特性，例如通用逼近能力和在适当条件下对对抗性示例的鲁棒性。然而，SMNN 在高维数据集中的应用存在一些瓶颈。首先，SMNN 预先计算了固定权重，并且迄今为止还没有定义 SMNN 训练过程，因此它们缺乏泛化能力。其次，SMNN 需要构建围绕输入数据集的凸多面体。在本文中，我们通过提出基于给定数据集的支持子集的 SMNN 训练程序来克服这些问题，并用基于超球面投影的方法替换凸多面体的构造。此外，本文还新介绍了SMNN的可解释能力和有效实施。